In wavelength division multiplexed (WDM) optical communication systems, a number of different optical carrier wavelengths are separately modulated with data to produce modulated optical signals. The modulated optical signals are combined into an aggregate signal and transmitted over an optical transmission path to a receiver. The receiver detects and demodulates the data.
One type of modulation that may be used in optical communication systems is phase shift keying (PSK). According to different variations of PSK, data is transmitted by modulating the phase of an optical wavelength such that the phase or phase transition of the optical wavelength represents symbols encoding one or more bits. In a binary phase-shift keying (BPSK) modulation scheme, for example, two phases may be used to represent 1 bit per symbol. In a quadrature phase-shift keying (QPSK) modulation scheme, four phases may be used to encode 2 bits per symbol. Other phase shift keying formats include differential phase shift keying (DPSK) formats and variations of PSK and DPSK formats, such as return-to-zero DPSK (RZ-DPSK) and phase division multiplexed QPSK (PDM-QPSK).
A modulation format, such as QPSK wherein multiple data bits are be encoded on a single transmitted symbol may be generally referred to as a multi-level modulation format. Multi-level modulation techniques have been used, for example, to allow increased transmission rates and decreased channel spacing, thereby increasing the spectral efficiency (SE) of each channel in a WDM system. One spectrally efficient multi-level modulation format is quadrature amplitude modulation (QAM). In a QAM signal, information is modulated using a combination of PSK and amplitude shift keying (ASK), for example, to encode multiple bits per symbol. A 16QAM modulation format may be used, for example, to encode 4 bits per symbol. Certain PSK modulation schemes (e.g., BPSK and QPSK) may be referred to as a level of QAM (e.g., 2QAM and 4QAM respectively).
Polarization multiplexing (POLMUX) may be implemented with a modulation format to double the spectral efficiency of the format. In a POLMUX format, two relatively orthogonal states of polarization of the optical carrier are separately modulated with data, e.g. using a QAM modulation format, and then combined for transmission. For example, in a POLMUX-QAM signal, orthogonal polarizations of the same optical carrier are modulated with different data streams using a QAM modulation format.
During transmission of the modulated signals over the transmission path, non-linearities in the path may introduce transmission impairments, such as chromatic dispersion (CD), polarization mode dispersion (PMD) and polarization dependent loss (PDL), into the signals. At the receiver, coherent detection may be used to detect the modulated optical signals. Digital signal processing (DSP) may be applied to the outputs of the coherent receiver to de-multiplex polarization multiplexed signals, compensate for transmission impairments such as PMD, PDL and other residual impairments, and demodulate the data.
In general, the DSP in the receiver may be configured to recover the transmitted signal by filtering the received signal with an adaptive filter having the inverse transfer function of the transmission path. Such an adaptive filter may be referred to as an equalizer. An ideal equalizer recovers the signals passed through the transmission path and completely removes the impairments imparted by the transmission path.
The coefficients of the equalizer, also referred to herein as “tap weights”, determine the transfer function of the equalizer. The tap weights are dynamically adjusted to minimize the error at the output of the equalizer. The error at the output of the equalizer is the difference between the actual output of the equalizer and the expected output. One way to acquire adjusted tap weights for the equalizer is to transmit a training sequence known by both the transmitter and receiver and to detect the impulse response of the transmission path from the training sequence. The receiver may then obtain the new tap weights by computing the inverse transfer function of the transmission path from the training sequence impulse response.
A second way to acquire adjusted tap weights for the equalizer is to start with initial values for the tap weights and design a cost function according to the characteristics of the received signal. The tap weights are continually adjusted by reducing the cost of the cost function until the error is minimized. When the error is minimized the equalizer is said to “converge.” Equalizers implementing this second approach to acquiring adjusted tap weights are referred to as “blind equalizers.” A blind equalizer is considered more effective than a non-blind equalizer since it continuously updates its transfer function to compensate for any changes in the transfer function of the transmission path without the need of synchronizing the training symbols.
A constant modulus algorithm (CMA) is one known algorithm used in blind-equalization. A CMA algorithm defines a cost function to estimate noise in a received signal. The higher the output (cost) of the cost function, the larger the noise and signal distortion in the received signal. The equalizer calculates an equalized signal by adding products of the received signal and the tap weights. After obtaining the equalized signal, the cost of the equalized signal is calculated using the cost function. The cost is used to adjust the tap weights of the equalizer. The equalizer then calculates a new equalized signal using the adjusted tap weights and obtains a new cost from the new equalized signal. The cost of the cost function is expected to be reduced by repeating this process.
A CMA is particularly useful in connection with PSK signals since it the equalized signal in a CMA converges on a constellation diagram with constellation points distributed over concentric circles. The CMA algorithm has been successfully implemented for POLMUX-QPSK signals for polarization demultiplexing and blind equalization. However, a CMA is not optimal for equalization of QAM signals since it produces a high mean square error (MSE) even when the equalizer is converged.
Several modifications to the CMA has been proposed, including a radius-directed decision-aided multimodulus algorithm (MMA), where a decision is made as to the ring of a constellation that a received symbol most likely belongs to and then the ring radius is adjusted. An example of an MMA is described in The Multimodulus Blind Equalization and Its Generalized Algorithms by Yang et al., published in IEEE Journal on Selected Areas in Communications, vol., No. 5, pp 997-1015, June 2002. One drawback of this type of MMA is that it relies on correct decisions of the ring radius. Also, the ring spacing in a 16 QAM signal, for example, can be smaller than the minimum symbol spacing. Accordingly, this type of MMA can produce large errors for low signal-to-noise ratio (SNR) and/or severe signal distortions.
Another challenge associated with known CMA and MMA equalizers is that they are insensitive to the carrier phase of the received optical signal. As such, an estimation of the carrier phase of the received signal is required when using CMA and MMA equalizers to ensure reliable demodulation of the data. A variety of approaches have been developed for performing carrier phase estimation (CPE) in the DSP of a receiver. These approaches are, however, complex and utilize the computing resources of the DSP.